From 754bbf7a25a8dda49b5d08ef0d0443bbf5af0e36 Mon Sep 17 00:00:00 2001 From: Craig Jennings Date: Sun, 7 Apr 2024 13:41:34 -0500 Subject: new repository --- devdocs/c/numeric%2Fcomplex%2Fcacosh.html | 80 +++++++++++++++++++++++++++++++ 1 file changed, 80 insertions(+) create mode 100644 devdocs/c/numeric%2Fcomplex%2Fcacosh.html (limited to 'devdocs/c/numeric%2Fcomplex%2Fcacosh.html') diff --git a/devdocs/c/numeric%2Fcomplex%2Fcacosh.html b/devdocs/c/numeric%2Fcomplex%2Fcacosh.html new file mode 100644 index 00000000..9da0e7a1 --- /dev/null +++ b/devdocs/c/numeric%2Fcomplex%2Fcacosh.html @@ -0,0 +1,80 @@ +

cacoshf, cacosh, cacoshl

Defined in header <complex.h> 
float complex       cacoshf( float complex z );
+		 (1) (since C99) 
double complex      cacosh( double complex z );
+		 (2) (since C99) 
long double complex cacoshl( long double complex z );
+		 (3) (since C99)
Defined in header <tgmath.h> 
#define acosh( z )
+		 (4) (since C99)
+1-3) Computes complex arc hyperbolic cosine of a complex value z with branch cut at values less than 1 along the real axis.
+4) Type-generic macro: If z has type long double complex, cacoshl is called. if z has type double complex, cacosh is called, if z has type float complex, cacoshf is called. If z is real or integer, then the macro invokes the corresponding real function (acoshf, acosh, acoshl). If z is imaginary, then the macro invokes the corresponding complex number version and the return type is complex.

Parameters

+ +
z - complex argument

Return value

The complex arc hyperbolic cosine of z in the interval [0; ∞) along the real axis and in the interval [−iπ; +iπ] along the imaginary axis.

+

Error handling and special values

Errors are reported consistent with math_errhandling

+

If the implementation supports IEEE floating-point arithmetic,

+

Notes

Although the C standard names this function "complex arc hyperbolic cosine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic cosine", and, less common, "complex area hyperbolic cosine".

+

Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment (-∞,+1) of the real axis.

+

The mathematical definition of the principal value of the inverse hyperbolic cosine is acosh z = ln(z + √z+1 √z-1) For any z, acosh(z) =

+√z-1/√1-z acos(z), or simply i acos(z) in the upper half of the complex plane.

Example

#include <stdio.h>
+#include <complex.h>
+ 
+int main(void)
+{
+    double complex z = cacosh(0.5);
+    printf("cacosh(+0.5+0i) = %f%+fi\n", creal(z), cimag(z));
+ 
+    double complex z2 = conj(0.5); // or cacosh(CMPLX(0.5, -0.0)) in C11
+    printf("cacosh(+0.5-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
+ 
+    // in upper half-plane, acosh(z) = i*acos(z) 
+    double complex z3 = casinh(1+I);
+    printf("casinh(1+1i) = %f%+fi\n", creal(z3), cimag(z3));
+    double complex z4 = I*casin(1+I);
+    printf("I*asin(1+1i) = %f%+fi\n", creal(z4), cimag(z4));
+}

Output:

+
cacosh(+0.5+0i) = 0.000000-1.047198i
+cacosh(+0.5-0i) (the other side of the cut) = 0.500000-0.000000i
+casinh(1+1i) = 1.061275+0.666239i
+I*asin(1+1i) = -1.061275+0.666239i

References

See also

+ + + + + +
+
(C99)(C99)(C99)
 computes the complex arc cosine
(function)
+
(C99)(C99)(C99)
 computes the complex arc hyperbolic sine
(function)
+
(C99)(C99)(C99)
 computes the complex arc hyperbolic tangent
(function)
+
(C99)(C99)(C99)
 computes the complex hyperbolic cosine
(function)
+
(C99)(C99)(C99)
 computes inverse hyperbolic cosine (\({\small\operatorname{arcosh}{x} }\)arcosh(x))
(function)
C++ documentation for acosh
+

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Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
+ https://en.cppreference.com/w/c/numeric/complex/cacosh +

+
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